An Improvement on Ranks of Explicit Tensors

TL;DR

This paper presents new explicit constructions of high-rank tensors with improved lower bounds, using recursive patterns and a partitioning theorem, advancing the understanding of tensor rank complexity.

Contribution

It introduces improved explicit tensor constructions with higher rank bounds, surpassing previous known bounds, based on a novel recursive pattern and a partitioning theorem.

Findings

Constructed tensors of rank at least 2n^k - O(n^{k-1})

Derived an n-shaped tensor with rank at least 2n^{r/2} - O(n^{r/2}-1) for odd r

Provided explicit tensor examples with improved rank bounds over prior work

Abstract

We give constructions of n^k x n^k x n tensors of rank at least 2n^k - O(n^(k-1)). As a corollary we obtain an [n]^r shaped tensor with rank at least 2n^(r/2) - O(n^(r/2)-1) when r is odd. The tensors are constructed from a simple recursive pattern, and the lower bounds are proven using a partitioning theorem developed by Brockett and Dobkin. These two bounds are improvements over the previous best-known explicit tensors that had ranks n^k and n^(r/2) respectively